Title: Unraveling the role of interactions in non-equilibrium transformations

Abstract: For arbitrary non-equilibrium transformations in complex systems, we show that the distance between the current state and a target state can be decomposed into two terms: one corresponding to an \textit{independent} estimate of the distance, and another corresponding to interactions, quantified using the relative mutual information between the variables. This decomposition is a special case of a more general decomposition involving successive orders of correlation or interactions among the system's degrees of freedom. To illustrate its practical significance, we study the thermalization of two optically trapped colloidal particles with hydrodynamic coupling. Our results show that, increasing pairwise interaction strength enhances the separation between the time-dependent non-equilibrium state and the final state, prolonging the non-equilibrium state's longevity. In more general setups, where it is possible to control the strength of different orders of interactions, our findings offer a way to disentangle their effects on the transformation process.